The effect of the tachocline on differential rotation in the Sun

In this paper, we present a model for the effects of the tachocline on the differential rotation in the solar convection zone. The mathematical technique relies on the assumption that entropy is nearly constant ("well-mixed") in isorotation surfaces both outside and within the tachocline. The resulting solutions exhibit nontrivial features that strikingly resemble the true tachocline isorotation contours in unexpected detail. This strengthens the mathematical premises of the theory. The observed rotation pattern in the tachocline shows strong quadrupolar structure, an important feature that is explicitly used in constructing our solutions. The tachocline is treated locally as an interior boundary layer of small but finite thickness, and an explicit global solution is then constructed. A dynamical link can thus be established between the internal jump in the angular velocity at the tachocline and the spread of angular velocities observed near the solar surface. In general, our results suggest that the bulk of the solar convection zone is in thermal wind balance, and that simple quadrupolar stresses, local in radius, mediate the tachocline transition from differential rotation to uniform rotation in the radiative interior.Comment: 20 Pages, 4 figures, to appear in MNRA

Gluing hyperconvex metric spaces

We investigate how to glue hyperconvex (or injective) metric spaces such that the resulting space remains hyperconvex. We give two new criteria, saying that on the one hand gluing along strongly convex subsets and on the other hand gluing along externally hyperconvex subsets leads to hyperconvex spaces. Furthermore, we show by an example that these two cases where gluing works are opposed and cannot be combined.Comment: 11 page

Global model of differential rotation in the Sun

The isorotation contours of the solar convective zone (SCZ) show three distinct morphologies, corresponding to two boundary layers (inner and outer), and the bulk of the interior. Previous work has shown that the thermal wind equation together with informal arguments on the nature of convection in a rotating fluid could be used to deduce the shape of the isorotation surfaces in the bulk of the SCZ with great fidelity, and that the tachocline contours could also be described by relatively simple phenomenology. In this paper, we show that the form of these surfaces can be understood more broadly as a mathematical consequence of the thermal wind equation and a narrow convective shell. The analysis does not yield the angular velocity function directly, an additional surface boundary condition is required. But much can already be deduced without constructing the entire rotation profile. The mathematics may be combined with dynamical arguments put forth in previous works to the mutual benefit of each. An important element of our approach is to regard the constant angular velocity surfaces as an independent coordinate variable for what is termed the "residual entropy," a quantity that plays a key role in the equation of thermal wind balance. The difference between the dynamics of the bulk of the SCZ and the tachocline is due to a different functional form of the residual entropy in each region. We develop a unified theory for the rotational behavior of both the SCZ and the tachocline, using the solutions for the characteristics of the thermal wind equation. These characteristics are identical to the isorotation contours in the bulk of the SCZ, but the two deviate in the tachocline. The outer layer may be treated, at least descriptively, by similar mathematical techniques, but this region probably does not obey thermal wind balance.Comment: 26 pages, 7 figures, accepted to MNRA

What do global p-modes tell us about banana cells?

We have calculated the effects of giant convection cells also know as sectoral rolls or banana cells, on p-mode splitting coefficients. We use the technique of quasi-degenerate perturbation theory formulated by Lavely & Ritzwoller in order to estimate the frequency shifts. A possible way of detecting giant cells is to look for even splitting coefficients of 'nearly degenerate' modes in the observational data since these modes have the largest shifts. We find that banana cells having an azimuthal wave number of 16 and maximum vertical velocity of 180 m/s cannot be ruled out from GONG data for even splitting coefficients.Comment: 7 pages 2 figures. To appear in Journal of Physics: Conference Series (JPCS) for GONG 2010 - SoHO 24: A new era of seismology of the Sun and solar-like star

On Differential Rotation and Convection in the Sun

We show that the differential rotation profile of the solar convection zone, apart from inner and outer boundary layers, can be reproduced with great accu- racy if the isorotation contours correspond to characteristics of the thermal wind equation. This requires that there be a formal quantitative relationship involving the entropy and the angular velocity. Earlier work has suggested that this could arise from magnetohydrodynamic stability constraints; here we argue that purely hydrodynamical processes could also lead to such a result. Of special importance to the hydrodynamical solution is the fact that the thermal wind equation is insensitive to radial entropy gradients. This allows a much more general class of solutions to fit the solar isorotation contours, beyond just those in which the entropy itself must be a function of the angular velocity. In particular, for this expanded class, the thermal wind solution of the solar rotation profile remains valid even when large radial entropy gradients are present. A clear and explicit example of this class of solution appears to be present in published numerical simulations of the solar convective zone. Though hydrodynamical in character, the theory is not sensitive to the presence of weak magnetic fields. Thus, the identification of solar isorotation contours with the characteristics of the thermal wind equation appears to be robust, accommodating, but by no means requiring, magnetic field dynamics.Comment: 16 pages, 2 figures. Accepted for publication in MNRA

Differential rotation and meridional circulation in global models of solar convection

In the outer envelope of the Sun and in other stars, differential rotation and meridional circulation are maintained via the redistribution of momentum and energy by convective motions. In order to properly capture such processes in a numerical model, the correct spherical geometry is essential. In this paper I review recent insights into the maintenance of mean flows in the solar interior obtained from high-resolution simulations of solar convection in rotating spherical shells. The Coriolis force induces a Reynolds stress which transports angular momentum equatorward and also yields latitudinal variations in the convective heat flux. Meridional circulations induced by baroclinicity and rotational shear further redistribute angular momentum and alter the mean stratification. This gives rise to a complex nonlinear interplay between turbulent convection, differential rotation, meridional circulation, and the mean specific entropy profile. I will describe how this drama plays out in our simulations as well as in solar and stellar convection zones.Comment: 5 pages, 2 figures (one color

The stability of stratified, rotating systems and the generation of vorticity in the Sun

We examine the linear behavior of three-dimensional Lagrangian displacements in a stratified, shearing background. The isentropic and iso-rotation surfaces of the equilibrium flow are assumed to be axisymmetric, but otherwise fully two-dimensional. Three-dimensional magnetic fields are included in the perturbation equations; however the equilibrium is assumed to be well-described by purely hydrodynamic forces. The model, in principle very general, is used to study the behavior of fluid displacements in an environment resembling the solar convection zone. Some very suggestive results emerge. All but high-latitude displacements align themselves with the observed surfaces of constant angular velocity. The tendency for the angular velocity to remain constant with depth in the bulk of the convective zone, together with other critical features of the rotation profile, emerge from little more than a visual inspection of the governing equation. In the absence of a background axial angular velocity gradient, displacements exhibit no poleward bias, suggesting that solar convection "plays-off" of prexisting shear rather than creates it. We argue that baroclinic vorticity of precisely the right order is generated at the radiative/convective zone boundary due to centrifugal distortion of equipotential surfaces that is not precisely followed by isothermal surfaces. If so, many features of the Sun's internal rotation become more clear, including: i) the general appearance of the tachocline; ii) the extension of differential rotation well into the radiative zone; iii) the abrupt change of morphology of convective zone isorotation surfaces; and iv) the inability of current numerical simulations to reproduce the solar rotation profile without imposed entropy boundary conditions.Comment: 30 pages, 2 figures. Accepted for publication in MNRA